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A search for Fibonacci-Wieferich and Wolstenholme primes


Authors: Richard J. McIntosh and Eric L. Roettger
Journal: Math. Comp. 76 (2007), 2087-2094
MSC (2000): Primary 11A07, 11A41, 11B39, 11Y99
DOI: https://doi.org/10.1090/S0025-5718-07-01955-2
Published electronically: April 17, 2007
MathSciNet review: 2336284
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Abstract: A prime $ p$ is called a Fibonacci-Wieferich prime if $ F_{p-({p\over5})}\equiv 0\pmod{p^2}$, where $ F_n$ is the $ n$th Fibonacci number. We report that there exist no such primes $ p<2\times10^{14}$. A prime $ p$ is called a Wolstenholme prime if $ {2p-1\choose p-1}\equiv 1\pmod {p^4}$. To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes $ p<10^9$. Wolstenholme, in 1862, proved that $ {2p-1\choose p-1}\equiv 1\pmod {p^3}$ for all primes $ p\ge 5$. It is estimated by a heuristic argument that the ``probability'' that $ p$ is Fibonacci-Wieferich (independently: that $ p$ is Wolstenholme) is about $ 1/p$. We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient $ F_{p-({p\over5})}/p$ modulo $ p$.


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Additional Information

Richard J. McIntosh
Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2
Email: mcintosh@math.uregina.ca

Eric L. Roettger
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: roettgee@math.ucalgary.ca

DOI: https://doi.org/10.1090/S0025-5718-07-01955-2
Keywords: Fibonacci number, Wieferich prime, Wall-Sun-Sun prime, Wolstenholme prime.
Received by editor(s): June 14, 2005
Received by editor(s) in revised form: May 19, 2006
Published electronically: April 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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